Magnetic and mechanical buckling: Modified Landau theory approach to study phase transitions in micromagnetic disks and compressed rods

Using the rigid magnetic vortex model, we develop a substantially modified Landau theory approach for analytically studying phase transitions between different spin arrangements in circular submicron magnetic dots subject to an in-plane externally-applied magnetic field. We introduce a novel order parameter: the inverse distance between the center of the circular dot and the vortex core. This order parameter is suitable for describing closed spin configurations such as curved or bent-spin structures and magnetic vortices. Depending on the radius and thickness of the dot as well as the exchange coupling, there are five different regimes for the magnetization reversal process when decreasing the in-plane magnetic field. The magnetization-reversal regimes obtained here cover practically all possible magnetization reversal processes. Moreover, we have derived the change of the dynamical response of the spins near the phase transitions and obtained a “critical slowing down” at the second order phase transition from the high-field parallel-spin state to the curved (C-shaped) spin phase. We predict a transition between the vortex and the parallel-spin state by quickly changing the magnetic field—providing the possibility to control the magnetic state of dots by changing either the value of the external magnetic field and/or its sweep rate. We study an illuminating mechanical analog (buckling instability) of the transition between the parallel-spin state and the curved spin state (i.e., a magnetic buckling transition). In analogy to the magnetic-disk case, we also develop a modified Landau theory for studying mechanical buckling instabilities of a compressed elastic rod embedded in an elastic medium. We show that the transition to a buckled state can be either first or second order depending on the ratio of the elasticity of the rod and the elasticity of the external medium. We derive the critical slowing down for the second-order mechanical buckling transition.

Figure 1

(a) Schematic diagram of a magnetic vortex sitting inside the dot and shifted from the dot center because of an applied in-plane external magnetic field $H$. The coordinate system is centered at the vortex core. The $z$-axis is perpendicular to the surface of the dot and towards the reader. (b) The so-called C-phase or bent spin configuration is modeled as a magnetic vortex sitting outside the dot. For this case $\left(s>1\right)$, the origin of the coordinate system is located at the dot center. The angle $\phi$ describes the possible rotation of the vortex around the dot center. The angle $\varphi$ and the $\varphi$-dependent distances ${\rho }_1\left(\varphi \right)$ and ${\rho }_2\left(\varphi \right)$ are used in Appendix in order to derive the exchange and Zeeman energies.

Figure 2

(Color online) (a–d) When increasing an externally applied force $f$ on a compressed bar, it eventually changes from a staight configuration (a) to a bent one (b) via a (first or second order) buckling transition. This Euler Buckling instability Ref. 38 is an example of spontaneous symmetry breaking (i.e., the bar can bend either facing left or right). We study this here as a phase transition. Here we also consider a magnetic analog of this mechanical spontaneous symmetry breaking transition. When decreasing the value of an externally applied magnetic field $H$, the spin arrangement inside a micromagnetic disk changes from a “straight” or “perfectly aligned” parallel-spin configuration (c) to a “bent” or C-shaped magnetization (d) with the spin configuration following a curved or bent shape. $1∕H$ plays the role of a generalized force. This is an example of a magnetic spontaneous symmetry breaking transition (e.g., the C-phase can face either to the left or to the right). This is studied here as a phase transition by expanding the magnetic energy of the dot. The order parameters ($M$ and ${\psi }_b$) versus “generalized forces” are schematically shown in (e,g). The effective response function, or susceptibility, for the standard ferromagnetic-paramagnetic phase transition and the buckling phase transition are compared in (f, h). (e) For zero externally applied magnetic field $H$, the magnetization $M$ is zero in the paramagnetic phase and starts to continuously increase in the ferromagnetic phase when lowering the temperature below $T_c$. As schematically shown in (e), an applied magnetic field $H\ne 0$ smears out the $M\left(T\right)$ phase transition, resulting in a nonzero magnetization in the paramagnetic phase. (f) The magnetic susceptibility $\chi =\partial M∕\partial H$ has a peak at $T_c$. (g) The transverse buckling amplitude ${\psi }_b\left(f\right)$ increases continuously when the compression force $f$ exceeds a critical value $f_c^{*}$, for small values of the elasticity $\alpha$ of the external medium. However, the continuous ${\psi }_b\left(f\right)$ transition becomes a discontinuous (first order) transition for higher values of the elasticity $\left(\alpha >{\alpha }_c\right)$. (h) The buckling of the rod becomes very sensitive to changes in the parameters when it is near $f_c^{*}$, exhibiting critical slowing down and a peaked effective susceptibility ${\chi }_b=\partial {\psi }_b∕\partial \alpha$. A closer analog of the external magnetic field $H$ is an external force $f_{\perp }$ perpendicular to the rod and applied near its center. Note that $f_{\perp }$ buckles the rod while $\alpha$ suppresses its buckling. More precisely, sufficiently large values of either $f_{\perp }$ or $f$ buckle the rod, when $\alpha$ is sufficiently weak. If $\alpha \to \infty$, then $f$ and/or $f_{\perp }$ will not be able to buckle the rod.

Figure 3

(Color online) (a) Phase diagram showing the domains of parameters for the different sequences of transitions among three stable and metastable spin configurations inside a micromagnetic disk. The parameters are the square of the reduced radius $R^2∕R_0^2$ versus the aspect ratio $\beta =L∕R$. Above the dotted top curve, the parallel spin state, which exists for high applied in-plane magnetic fields, discontinuously transforms to a magnetic vortex sitting inside the disk. This is because the large radius of the microdisk can easily accommodate a vortex inside. For parameter values between the dotted and continuous curves (e.g., for smaller disk radius at a fixed $\beta$), a second-order phase transition from the parallel spin configuration to the $\mathrm{C}$-phase occurs first. Upon further lowering the in-plane field, the $\mathrm{C}$ phase abruptly transforms to a vortex sitting inside the disk. Between the dashed and continuous curves, the $\mathrm{C}$-phase survives down to $h=H∕M_s=0$, and rotates as a whole when the magnetic field changes its polarity. This rotation costs little or no energy and it is a Goldstone mode (Refs. 61, 62, 63, 64, 65, 66). Below the dashed curve, the magnetization reversal process proceeds via a rotation of the parallel-spin state as a whole at $h=0$. Even though the vortex state does not contribute to the magnetization reversal process below the continuous curve, the magnetic vortex is stable or metastable at low magnetic fields above the continuous curve located at the bottom of the diagram. In practice, the stable or metastable vortex states below the continuous curve cannot be reached besides at high temperatures or when $H$ changes suddenly. Below this continuous bottom curve, the magnetic vortex does not correspond to an energy minimum and it is unstable for any value of $h$. (b) Phase diagram plotted on the plane [$\text{magnetic field}={\left(\text{generalized force}\right)}^{-1}=h=H∕M_s$$$, $\text{reduced}\text{radius}=R∕R_0$]. This shows magnetization reversal processes for a fixed aspect ratio $\beta =L∕R=1$ when lowering the applied magnetic field $h$ from a high value $h>h_c$. Dashed (solid) lines correspond to first-(second-) order phase transitions. The short black segment at the top left corner corresponds to Goldstone modes, where the magnetization can be rotated with zero or little energy cost. For the case of increasing $h$ from a high negative value $h<-h_c$, the diagram is inverted with respect to the $h=0$ axis. In order to construct this diagram, we use Eqs. (18, B4) and the criterion $\psi =1$ for penetrating a magnetic vortex (transition from the C-phase to the vortex state).

Figure 4

(Color online) The evolution of the dependence of the total energy $w\left(\psi \right)$ when lowering the applied in-plane magnetic field $h\equiv H∕M_s$, for the parameters chosen at point A of the diagram in Fig. 3a. For high $h$ (e.g., $h=5$) there is only one minimum of the energy $w$ corresponding to the parallel spin state (the bottom dashed line). For lower values of the applied field, a metastable energy minimum associated with a vortex inside the disk appears and then deepens when further lowering $h$ (the continuous line). Finally, for even lower values of $h$, the minimum energy corresponding to the parallel spin state disappears (the top dotted line) and the spin arrangement discontinuously changes to a vortex state. The corresponding magnetization loop is shown in the right bottom corner.

Figure 5

(Color online) Total magnetic energy $w$ versus order parameter $\psi$ when lowering the in-plane magnetic field $h$ for the parameters chosen at point B on the diagram in Fig. 3a. For relatively high magnetic fields, the observed change of $w\left(\psi \right)$ is similar to the one shown in Fig. 4 (the dashed curve at the bottom, $h=2$, and the middle continuous curve, $h=1$). However, the minimum in $w\left(\psi \right)$, originally corresponding to the parallel spin state, $\psi =0$, starts to shift to the right resulting in a second-order phase transition to the $\mathrm{C}$-phase. In this case, the vortex begins to “continuously penetrate” the disk from infinity (see the left portion of the top dotted curve which is inside the left inset). The metastable $\mathrm{C}$-phase $\left(\psi \ne 0\right)$ shown there continuously evolves from the parallel state at $\psi =0$. The minimum in $w\left(\psi \right)$ corresponding to the $\mathrm{C}$-phase survives even at $h=0$. For very low negative fields, the C-phase rotates as a whole (Goldstone mode corresponding to a zero-energy rotation of $\mathit{M}$) and the vortex state does not contribute to the reversal magnetization process even though it has a minimum energy at low fields. The corresponding magnetization loop with the virgin magnetization curve related to the vortex state is shown in the right bottom corner. The dashed arrow shows how one could access the vortex state by a sudden drop (or jump) of the external magnetic field.

Figure 6

(Color online) Main panel: dependence of the total energy $w\left(\psi \right)$ when lowering the in-plane magnetic field $h$ for the parameters chosen at point C on the diagram in Fig. 3a. For high and positive $h$ (e.g., $h=2$, 0.8), the behavior of $w\left(\psi \right)$ is similar to the one seen in Figs. 4, 5 (the bottom dashed and the continuous curve). However, the parallel spin state is metastable even at zero magnetic field (the dotted line). For negative values of the external field, the parallel-spin state rotates as a whole (Goldstone mode). If the magnetic field suddenly drops to the negative value $h=-0.6$, then a gradient of the energy towards the vortex state appears (the top dotted-dashed curve). As shown in the top right inset, there is no energy minimum corresponding to the vortex state for the parameters corresponding to point D in Fig. 3a.

Figure 7

(Color online) Phase diagram for the mechanical buckling instability. The parameters shown are the normalized force $=f{l}^2∕\left({\pi }^{2}IE\right)$ versus the ratio of the medium/rod elasticities $\alpha l^4∕\left({\pi }^{4}IE\right)$. Solid segments correspond to second-order phase transitions from the straight rod state to the buckled state, while the first-order phase transitions are indicated by the dashed lines. The different buckling modes, with different undulation numbers $n^{*}$, are sketched in the corresponding phase domains. The black vertical dotted segment around $\alpha l^4∕\left({\pi }^{4}IE\right)\approx 4$ corresponds to a first-order transition between the $n^{*}=1$ (C-phase-like) and the $n^{*}=2$ (S-phase-like). The transition from the straight rod phase to the $n^{*}=1$ buckled phase is the mechanical analog of the magnetic phase transition between the straight and bent spin-configuration phases: this diagram corresponds to the one shown in Fig. 3b. The transition to the $n^{*}=2$ buckled rod state is the mechanical analog of the magnetic transition to the S phase in microdisks.